Rucete ✏ AP Physics C In a Nutshell
7. Rotation II: Inertia, Equilibrium, and Combined Rotation/Translation
This chapter covers the calculation of rotational inertia for complex shapes, the parallel axis theorem, angular momentum, conservation of angular momentum, rolling without slipping, and static equilibrium of extended objects. It extends rotational dynamics to real-world systems involving both rotation and translation.
Relationship Between Torque and Angular Momentum
• Net external torque equals the time rate of change of angular momentum:
τnet = dL/dt
• If τnet = 0, angular momentum is conserved.
• Internal forces and torques cancel out when calculating the net external torque on a system.
Conservation of Angular Momentum
• Angular momentum is conserved when no external torques act on a system.
• Applies to rotating systems like tops, figure skaters pulling arms in, or collapsing stars.
• If I changes (e.g., arms pulled inward), ω adjusts to conserve L = Iω.
Rolling Without Slipping
• Condition: vCM = Rω
• Combined translation of the center of mass and rotation about the center.
• Kinetic energy: KEtotal = ½MvCM² + ½ICMω²
• Can model rolling as pure rotation about the point of contact with the ground.
Force Required for Rolling Without Slipping
• No net torque is needed for constant-speed rolling without slipping.
• When external forces are applied (e.g., pushing a wheel), static friction provides the torque needed for rolling without slipping.
• If static friction is insufficient, slipping occurs (rolling with sliding).
Static Equilibrium for Extended Objects
• Two conditions for static equilibrium:
– ΣF = 0 (no linear acceleration)
– Στ = 0 (no angular acceleration)
• Choosing a clever pivot point can eliminate unknown forces from torque equations.
• The gravitational force can be treated as acting at the center of mass.
Solving Static Equilibrium Problems
• Draw a complete free-body diagram.
• Resolve forces into components along convenient axes.
• Apply ΣFx = 0 and ΣFy = 0.
• Choose an axis and apply Στ = 0 (torque equilibrium).
• Solve the resulting system of equations for unknowns.
In a Nutshell
Rotational dynamics mirror linear motion but involve torque and angular momentum. Objects in static equilibrium must satisfy both force and torque balance. Rolling motion combines translation and rotation, and the parallel axis theorem simplifies inertia calculations. Understanding these principles enables analysis of complex mechanical systems, from spinning tops to balancing ladders.